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| HAL: hal-00715060, version 1 |
| arXiv: 1207.1579 |
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| Betti numbers of random real hypersurfaces and determinants of random symmetric matrices |
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| Damien Gayet 1Jean-Yves Welschinger 1 |
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| (2012-07-06) |
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| We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the ran- dom real hypersurfaces. |
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| 1: | Institut Camille Jordan (ICJ) |
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CNRS : UMR5208 – Université Claude Bernard - Lyon I – Ecole Centrale de Lyon – Institut National des Sciences Appliquées (INSA) - Lyon |
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| Subject | : | Mathematics/Algebraic Geometry Mathematics/Complex Variables Mathematics/Probability |
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| Real projective manifold – ample line bundle – random matrix – random polynomial |
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| hal-00715060, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00715060 | |
| oai:hal.archives-ouvertes.fr:hal-00715060 | |
| From: Damien Gayet | |
| Submitted on: Friday, 6 July 2012 11:42:53 | |
| Updated on: Friday, 6 July 2012 13:59:12 | |
